**Statistical Methods in Financial Markets**

An important complement to differential equation analysis is
statistical time series methodology. The
advantage of time series is that the model is chosen in a prescribed deductive
manner which provides further evidence of the fundamental forces that govern
price dynamics beyond valuation. In
particular the Box-Jenkins methodology for selecting the appropriate ARIMA
model. This procedure then selects from
a large collection of models that include a simple random walk, a pure momentum
model and more complicated models.

Thus, an ARIMA analysis of a data set has the potential to
determine, for example, whether today price derivative predicts tomorrow's
price derivative. Acquiring an
understanding of markets is greatly complicated by ''noise'' or random events
that affect valuation, so that any deterministic forces are difficult to
detect.

One way to extract the deterministic forces from the random
events is to utilize ratios of pairs of stock that are identical in fundamental
value. An ARIMA model extracted from
this ratio data can be compared to others and to data compiled from the
laboratory experiments. In particular
this provides a quantitative link that is very convincing.

The ARIMA methodology can also be
utilized in deriving the terms of the differential equations in an empirical
way.